Minds On

Types of triangles

For each description, select the corresponding triangle.

For each description, select the corresponding triangle.

Press 'Let's check!' to reveal the types of triangles.

The chart categorizes the different types of triangles by side and by angle. The first column is labelled “By side” and contains 3 different triangles. The first is the equilateral triangle, which has three equal sides, and each side has a hatch mark. The next triangle is the isosceles triangle, which has two equal sides, so the two sides that are the same have hatch marks. The last triangle is the scalene triangle, which has no equal sides, so all sides are different lengths. The second column is labelled “By angle” and contains 3 different triangles. The first triangle is the acute triangle which has three angles that are less than 90 degrees. The next triangle is the right triangle, which has one right angle that is equal to 90 degrees. In this triangle, there is a small square in one corner of the triangle, which signifies that this angle is 90 degrees. The last triangle is the obtuse triangle which has one angle that is greater than 90 degrees.

Creating triangles

Student Success

Think-Pair-Share

Now let's create and explore triangles with a partner, if possible.

If you would like, you can complete the next activity using TVO Mathify or a method of your choice. You can also use your notebook or the following fillable and printable document.

Press the ‘TVO Mathify’ button to access this resource and the ‘Activity’ button for your note-taking document.

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Note to teachers: See your teacher guide for collaboration tools, ideas and suggestions.

Action

Task 1: Sum of interior angles in a triangle

The sum of the interior angles in a triangle is always 180°.

A triangle has three angles labelled A, B, and C. The sum of angle A, angle B, and angle C is equal to 180 degrees. The sum of the angles in a triangle is always 180 degrees.

Unknown angles

Now, let's determine the unknown angles on some triangles.

Select the correct answer, then press "Check Answer" to see how you did.

Press each of the following triangles to reveal the unknown angles.

Triangle A B C:

  • Angle A = 100 °
  • Angle B = 25 °
  • Angle C = 55 °

Since the sum of the interior angles in a triangle is 180 ° , we can do 180 ° − 100 ° − 55 ° = 25 °

Therefore, Angle B is 25 ° .

Triangle J K L:

  • Angle J = 71 °
  • Angle K = 53 °
  • Angle L = 56 °

Since the sum of the interior angles in a triangle is 180 ° , we can do 180 ° − 71 ° − 56 ° = 53 °

Therefore, Angle K is 53 ° .

Task 2: Dividing polygons

We can determine the angle sum of any polygon by dividing it into triangles.

The number of triangles the shape can be divided into with line(s) coming from only one vertex, will determine the total sum of the interior angles of the polygon.

Student Tips

Polygons and angles

Reminder: The sum of angles in a triangle is equal to 180 °

Consider:

  • How many triangles can you divide a quadrilateral into using the vertices only?
  • What is the sum of the angles in a quadrilateral?

Let's explore the following example.

This is a quadrilateral that can be divided into two triangles.

A square is divided in half with a dotted line. The line is connected from the top right corner to the bottom left corner to form two triangles. For each triangle, the sum of interior angles is 180 degrees. The equation for the sum of the angles in the quadrilateral is 2 times 180 degrees, which equals to 360 degrees.

The sum of the interior angles of each triangle is 180 ° .

Therefore, we can determine the sum of the interior angles of this quadrilateral:

Sum of interior angles of triangle 1 + sum of interior angles of triangle 2 = sum of all angles in this quadrilateral

Or

180 ° + 180 ° = 360 °

How many triangles?

Let's explore how many triangles are required to make up some other polygons!

If you would like, you can complete the next activity using TVO Mathify or a method of your choice. You can also use your notebook or the following fillable and printable document.

Press the ‘TVO Mathify’ button to access this resource and the ‘Activity’ button for your note-taking document.

TVO Mathify (Opens in a new window)Activity(Open PDF in a new window)

Press each of the following titles to check the number of triangles in the following polygons.

A pentagon can be divided into three triangles. The interior angles for each triangle has a sum of 180 °

In other words, 3 × 180 ° = 540 °

The pentagon has five corners and sides and is divided into three triangles. The bottom left vertex has two dotted lines extending to opposite vertexes to create three triangles. For each triangle, the sum of interior angles is 180 degrees. The equation for the sum of the angles in the polygon is 3 times 180 degrees, which equals to 540 degrees.

A hexagon can be divided into four triangles. The interior angles for each triangle has a sum of 180 °

In other words, 4 × 180 ° = 720 °

The hexagon has six corners and sides and is divided into four triangles. The top left vertex has three dotted lines extending to opposite vertexes to create four triangles. For each triangle, the sum of interior angles is 180 degrees. The equation for the sum of the angle in the polygon is 4 times 180 degrees, which equals to 720 degrees.

Task 3: Calculating angles of a polygon

Brainstorm

Let's think!

Reflect on the previous task to guide your thinking with the following questions:

  • What is the relation between the number of sides the polygon has, and the number of triangles it can be divided into?
  • What did you notice between the number of sides of the polygon and the number of triangles you were able to divide it into?

If the polygon has n number of sides, the number of triangles it can be divided into will be n – 2

Let's prove this by examining the quadrilateral from the previous task:

A square is divided in half by a dotted line from the top right corner to the bottom left corner to form two triangles. For each triangle, the sum of interior angles is 180 degrees. The equation for the sum of the angles in the quadrilateral is 2 times 180 degrees, which equals to 360 degrees.

This quadrilateral has 4 sides and is divided into two triangles.

If a polygon has n number of sides, then the number of triangles it can be divided into is n − 2.

n − 2 = number of triangles it can be divided into.

4 sides of a quadrilateral − 2 = number of triangles it can divided into.

4 sides of a quadrilateral − 2 = 2 triangles.

Therefore, the relationship between the number of sides and triangles on the inside is n − 2.

Press 'Let's check!' to reveal the number of triangles in a pentagon and a hexagon.

If we apply the equation to the pentagon and hexagon:

  • The pentagon had 5 sides and can be divided into 3 triangles.
  • The hexagon had 6 sides and can be divided into 4 triangles.

Extend and calculate

We can extend this formula to calculate the sum of the interior angles of a polygon:

(n − 2) × 180° = sum of the interior angles of a polygon

If you would like, you can complete the next activity using TVO Mathify or a method of your choice. You can also use your notebook or the following fillable and printable document.

Press the ‘TVO Mathify’ button to access this resource and the ‘Activity’ button for your note-taking document.

TVO Mathify (Opens in a new window) Activity(Open PDF in a new window)

Consolidation

Exploring polygons and angles

If you would like, you can complete the next activity using TVO Mathify or a method of your choice. You can also use your notebook or the following fillable and printable document.

Press the ‘TVOMathify’ button to access this resource and the ‘Activity’ button for your note-taking document.

TVO Mathify (Opens in a new window)Activity(Open PDF in a new window)

Reflection

As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.

I feel...

Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text, or writing tool.

When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.

Connect with a TVO Mathify tutor

Think of TVO Mathify as your own personalized math coach, here to support your learning at home. Press ‘TVO Mathify’ to connect with an Ontario Certified Teacher math tutor of your choice. You will need a TVO Mathify login to access this resource.

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